Toda brackets and congruences of modular forms

TL;DR

This paper explores the connection between Toda brackets and modular form congruences, extending classical invariants to higher chromatic levels and providing new formulas for the $f$-invariant.

Contribution

It generalizes Adams' formulas for the $e$-invariant to the chromatic second filtration by determining the $f$-invariant of Toda brackets.

Findings

Determined the $f$-invariant of Toda brackets.

Generalized classical $e$-invariant formulas.

Extended understanding of modular form congruences.

Abstract

This paper investigates the relation between Toda brackets and congruences of modular forms. It determines the $f$-invariant of Toda brackets and thereby generalizes the formulas of J.F.\ Adams for the classical $e$-invariant to the chromatic second filtration.